Oscillation Of Higher-order Differential Equations And Second-order Difference Equations

With the increasing developing of science and technology, there are many problems relating to differential equation derived from lots of practical applica-tions, such as whether all of its solutions are oscillation or not. The oscillation of ordinary differential equation is one of important branch of differential equations. It has deep physical background and mathematical models. In this dissertation, we employ a generalized Riccati transformation, estimations of inequalities, in-tegral average technique and monotonicity of a function to investigate the oscil-lation problems for two classes of higher-order differential equations, we obtain some meaning fully new results. Besides, by employing functions H(m,n) and h(m,n), generalized Riccati transformation investigate the oscillation criteria of a class of second-order nonlinear neutral delay difference equation shall further the investigation and improve the main results.The thesis is divided into four sections according to the contents.In chapter 1, we introduce the main contents of this paper.In chapter 2, new oscillation criteria for high-order neutral delay differential equationwhere z(t)= u(t)+p(t)u(t-σ), a> 0,αi> 0(i= 1,2,3,..., m) are established under the conditions orby employing generalized Riccati transformation, integral average technique and the estimations of ineqalities. An example illustrating the application of our results is also provided. In chapter 3, we study the oscillation of the higher-order nonlinear differential equation where Y(t)= u(t)+p(t)u(t-τ),τ≥0, nis even. By employing a new functionΦ(t, s,l), integral average technique and generalized Riccati transformation, we obtain new oscillation criteria, which extend and improve the known results.In chapter 4, we investigate the oscillation criteria of second-order nonlinear neutral delay difference equationwe mainly by employing functions H(m,n) and h(m,n), generalized Riccati transformation shall further the investigation and improve the main results of S.H.Saker[34], we obtained several new oscillation criteria at the end of this sec-tion.